Figure 1 : Eddies in the PBL
The turbulent edies
are carried upward on the envelopes of the large eddies. Buoyant turbulent
kinetic energy generation by MFC occurs in the environment of the
turbulent eddies and gives rise to the formation of microscale capping
inversion ( MCI
) layer on the large eddy envelope. The MCI is
seen as the rising inversion of the day time PBL in echosonde and
Townsend ( 1956 ) has derived the following relationship between the large and turbulent eddy root mean square ( r. m. s ) circulation speeds W and w* and their respective radii R and r as follows
W is approximately equal to 0.25 for R/r = 10 where R/r is the length scale ratio z . Therefore upward turbulent vetical velocity production at the planetary surface gives rise to upward propagating large eddies or gravity ( buoyancy ) waves.
The MCI is a region of vertical mixing since the turbulent eddy fluctuations mix overlying environmental into the large eddy volume. The fractional volume dilution rate k of the total large eddy volume by vetical mixing is derived as ( Mary Selvam et al., 1985a )
where w* is the turbulent vertical velocity production by MFC and dW is the corresponding increase in the large eddy circulation speed. The value of k is greater than 0.5 for length scale ratio z less than 10 . Though a continuous spectrum of eddies propagate outward from the planetary surface , only those eddies with length scale ratio greater than or equal to 10 can exist as discrete, identifiable, semipermanent entities in the PBL since dilution by vertical mixing erases their ( large eddy ) signature for smaller scale ratios. In summary, the PBL contains a semipermanent hierarchical system of eddies consisting of the convective, meso-, synoptic and planetary scales which evolve basically from the dominant turbulence scale at successive decadic scale range intervals and is manifested as mesoscale cloud clusters and cloud rows in global synoptic weather systems. Enhanced condensation inside clouds amplifies the myriads of turbulent eddies and gives rise to cloud top gravity oscillations. Cloud water condensation in the innumerable turbulent eddies is responsible for the observed cauliflower-like surface granularity of the cumulus cloud. The physical mechanism of growth of the atmospheric buoyancy ( gravity ) waves from turbulent buoyant energy production is analogous to the conditional instability of the second kind ( CISK ) mechanism ( Holton, 1979 ) where hurricane systems are postulated to derive their energy from convective scale cloud condensation. Also there is an inherent two-way energy feedback mechanism in the hierarchical system of eddies discussed in this paper and given in Equation 1 which is a statement of the law of conservation of energy, selfsimilarity and self-consistency in atmospheric processes. The kinetic energy E of unit volume of the atmospheric eddy of frequency n may be shown to be equal to Hn where H is the instantaneous spin angular momentum of unit volume of planetary scale eddy about the earth's axis. It is further shown ( Mary Selvam, 1986 ) that the eddy energy spectrum gives the probability density distribution of the eddy field. Thus, the physical laws governing eddy dynamics in the macroscopic planetay atmosphere is analogous to the quantum mechanical laws of the sub-atomic space. Therefore the mesoscale cloud clusters are a visible macroscale manifestation of the universal quantum mechanical nature of the energy stucture of natural phenomena. The full continuum of atmospheric eddies exist as a unified whole in time and space and contribute to the manifested atmospheric phenomena in the global planetary atmosphere and such a concept is similar to the bootstrap theory of Chew ( 1968 ) and the theory of implicate order envisaged by Bohm ( 1951 ).
The relationship between the size ( R ) , time period ( T ), circulation velocity ( W ) and energy ( E ) scales of the convective ( c ) , meso- ( m ), synoptic ( s ) and planetary ( p ) scale atmospheric eddy system to the basic turbulence scale ( t ) is derived from Equation 1 and is given below.
R : Rt = r : 10r : 10
: 10 3r : 10 4r
T : tt = t : 40t : 40 2t : 40 3t : 40 4t
W : Wt = w* : 0.25w* : 0.25 2w* : 0.25 3w* : 0.25 4w*
E : Et = e : 62.5 e : 62.5 2e : 62.5 3e : 62.5 4e
The globally observed quasi-biennial oscillation ( QBO ) and the 20-year cycle in weather patterns ( Burroughs, 1986 ) may possibly result respectively from the fundamental semi-diurnal atmospheric oscillation ( QBO ~ 12 hrs x 402 ) and the 5 - minutes oscillations of the sun's atmosphere ( 20 years ~ 5 minutes x 404 ) ( Equation 3 ). Such a process is analogous to anti-Stokes laser emission triggered by laser pump.
This is the well known
logarithmic wind profile relationship in the surface ABL and the
Karman's constant k is now defined by the new theory
as representing the fractional volume dilution rate of the large eddy by
vertical mixing due to turbulent eddy fluctuations for a scale ratio of
. Further, the theory states that the logarithmic wind profile relationship
prevails throughout the PBL , the absolute value of W
being determined solely by MFC in the dominant turbulent eddy. Logarithmic
spiral airflow tracks ( vortices ) are therefore associated with vortex
roll circulations and is consistent with observations ( Hauf, 1985 ). The
atmospheric eddy continuum therefore consists of a hierarchical system
of vortices within vortices.
The parcels of air rising up from the surface in the updraft regions of large eddy circulations get diluted by vertical mixing and only a fraction f reaches the normalised height z and is given by ( Mary Selvam et al ., 1985 ).
W = w* f z
Since the eddy energy is derived from microscale fractional condensation on hygroscopic nuclei in the troposphere the eddy enegy spectrum is dependent on the atmospheric nuclei size spectrum. The observed atmospheric nuclei spectra follow the Junge aerosol size spectrum ( Pruppacher and Klett, 1978 ) and it is shown that the observed aerosol size spectrum is a natural consequence of the atmospheric eddy continuum ( Ramachandra Murty et al ., 1985 ). The aerosol size spectrum has a computed maximum spectral slope of -4.8 in the maximum size region.
Figure 2 : Universal pressure and wind anomaly patterns in synoptic eddy systems
Decrease in r results in
increase in absolute values of the pressure and wind gradients.
The above physical mechanism envisaged for atmospheric eddy growth postulates the co-existence of large and small eddies in a hierarchical system with ordered dynamical local-global mutual energy exchange and therefore all energy systems are inherently non-local. It is shown that such a concept leads to the universally observed normal distribution characteristics in natural phenomena. Also, Lovejoy and Schertzer( 1986 ) have established the fractal characteristics of rain areas and rainfall in association with the observed scale invariant characteristics of the atmospheric eddy energy spectrum. The regions of eddy continuum energy enhancement are conventionally associated with ordered chaos. The universal route to chaos, namely, period doubling, intermittency and scale invariant eddy energy structure ( Harrison and Biswas, 1986 ) is commonly observed in all natural growth phenomena ( Fairbairn , 1986 ) and therefore is an inherent characteristic of the quantum mechanical nature for the eddy energy structure manifestation in natural phenomena. Since eddy circulation inherently consists of opposite directions of motion in association with different manifested phenomena, the apparent quantisation of energy in nature occurs. Eddy motions or vibrations in the PBL are responsible for the observed normal distribution characteristics of thermodynamical parameters as explained in the following. The eddy energy propagates by the inherent property of inertia of the medium and therefore w* , w*2 , w*3 and w*4 respectively represent the inertia, force, angular momentum and spin angular momentum of the medium caused by the eddy motion initiated by the turbulence scale acceleration w* per second. Therefore, the mean ( w ), variance ( s2 ), skewness and kurtosis are respectively given by w* , w*2 , w*3 and w*4 . The moment coefficient of skewness is equal to zero. The moment coefficient of kurtosis is equal to three and represents a factor of three increase in the spin angular momentum for the inherent period doubling process of large eddy growth in the MCI . Therefore the normal distribution characteristics conventionally attributed to random chance are in reality the deterministic laws of eddy growth by the period doubling process and represent the implicit order in natural phenomena. Signal and noise exist in each other in nature since signal is but the integrated mean of the conventional background random noise over large space and time scales. Further, the following fundamental relations in statistics and mathematics may be obtained from the physical concept of the eddy growth mechanism as follows. The standard equation relatining population and sample standard deviation follows from Equation 1, namely, W22 = W12 /n where n = z2 / z1 , and z1, z2 are the respective scale ratios with respect to the turbulence scale for two large eddy circulations of root mean square circulation speeds W1 and W2 respectively. The function f also gives the angular displacement of the particle trajectory on the large eddy envelope for successive radial growth steps r and it may be shown that f = p = 22/7 = circumference/diameter for a complete large eddy circulation of scale ratio 10 . The hierarchical growth of large eddies from turbulence scale is basically by a period doubling process in the MCI where incremental growth occurs in length steps equal to r . The layer MCI is thus a region of chaos. The unique particle trajectory design of concentric circles in the field of chaos has been named the strange attractor and results from the inherent hierarchy of the continuum vortex roll circulations. For example, the strange attractor design is seen in the MCI where maximum aerosol concentration occurs with a layered fine structure representing the component turbulent eddies and has been observed commonly in the stratospheric Junge aerosol layer, the arctic haze ( Radke et al ., 1984 ) and more spectacularly in the planetary rings of Jupiter, Saturn and Uranus ( Michel, 1985 ) indicating dusty debris filled atmosphere and violent equatorial convective dynamics for the major planets. Particles in the planetary rings may be shown to follow the relation R3 / T2 = constant from Equations 1 and 2 and is therefore consistent with Kepler's third law of planetary motion.
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